A
new problem of social choice has attracted the attention of scholars in
law, economics, political science, philosophy and computer science. How
can a group of individuals aggregate the group members' individual
judgments on some interconnected propositions into corresponding collective judgments on these
propositions? Such aggregation problems occur in many
different collective decision-making bodies, for example in committees,
legislatures, judiciaries and expert panels.

Judgment aggregation is distinct from the more familiar problem of
preference aggregation. But just as preference aggregation is illustrated
by a paradox (Condorcet's paradox of cyclical majority
preferences), so judgment aggregation is also illustrated by a paradox:
the "discursive dilemma" or "doctrinal paradox".

This
page provides a bibliography of online and published research on this paradox
and on judgment aggregation more generally.

Suppose
that a three-member court has to make a judgment on whether a defendant is liable
for a breach of contract. According to legal doctrine, the defendant is liable
(proposition R) if and
only if the defendant did some action X (proposition
P) and the defendant had a contractual obligation
not to do action X (proposition Q). Thus legal doctrine requires R<->(P&Q).
Suppose that the individual judgments of the three judges are as in table
1.

All
three judges accept the rule R<->(P&Q).
Further, judge 1 accepts both P and Q and, by implication, R.
Judges 2 and 3 each accept only one of P or Q and, by
implication, they both reject R. If the court applies majority
voting on each proposition (including on R<->(P&Q)),
it faces a paradoxical outcome. A majority accepts P, a majority
accepts Q, a majority (unanimity) accepts R<->(P&Q),
and yet a majority rejects R.

In
earlier presentations of the problem under the name "doctrinal
paradox", the logical connection rule R<->(P&Q)
was not considered as a proposition on which the court explicitly makes a
judgment by majority voting, but it was held fixed in the background as an
exogenous constraint or "legal doctrine". This restriction was
given up in more recent presentations of the problem under the name "discursive dilemma"
(Pettit 2001b; List
and Pettit 2002).

Propositionwise
majority voting thus produces an inconsistent collective set of judgments,
namely the set {P, Q, (R<->(P&Q)),
not-R}
(corresponding to the last row of table 1). This set is inconsistent in
the standard sense of propositional logic: There exists no assignment
of truth-values to propositions P, Q and R that makes all the propositions in the set simultaneously true.
This
outcome occurs although the sets of judgments of individual judges
(corresponding to the first three rows of table 1) are all consistent.

For
a recent discussion of the paradox by Kornhauser and Sager, see Kornhauser
and Sager (2004, cited below); for a response, see List
and Pettit (2005). For a discussion of the discursive dilemma from a
social epistemology perspective, see Goldman
(2004) and List (2005).

the
premise- and conclusion-based procedures

Premise-based
and conclusion-based procedures of decision-making have been proposed as
possible escape-routes from the paradox. These procedures interpret
propositions P and Q as premises, (R<->(P&Q))
as a rule of inference, and R as a conclusion.

According
to the premise-based procedure, the group applies majority voting
on each premise (i.e. propositions P and Q), but not
on the conclusion (i.e. proposition R), and derives the collective
judgment on that conclusion (i.e. R) on the basis of the
appropriate logical connection rule (i.e. proposition R<->(P&Q)).
Under this procedure, the group effectively ignores the majority verdict
on the conclusion. In
table 1, the premise-based procedure leads to the collective acceptance of
the conclusion.

According
to the conclusion-based procedure, the group applies majority
voting only on the conclusion (i.e. R), but not on the premises
(i.e. P and Q), ignoring the majority
verdicts on them. In table 1, the
conclusion-based procedure leads to the collective rejection of the
conclusion.

Thus
the premise-based and conclusion-based procedures may produce different outcomes.

For
a discussion of the premise- and conclusion-based procedures, see Pettit
(2001b) (a deliberative democracy and republican perspective), Bovens
and Rabinowicz (2003, 2004) and List
(2005) (an epistemic
perspective, focusing on the truth-tracking capacities of the two
procedures), Chapman
(2002) (a common law
perspective), Dietrich and List
(2004a) (a discussion of the strategic
incentives created by the two procedures).

an
impossibility theorem and more general developments

It
can be shown that the paradox is not just an artefact of majority voting, but that it
illustrates a more general impossibility theorem.

A
(judgment) aggregation procedure is a function which takes as
its input a profile of individual sets of judgments across the members of a group, and which produces as its output a
collective set of judgments.

The sets of
judgments of each
individual are assumed to satisfy certain consistency conditions (completeness, consistency and deductive closure).

For
the following first impossibility result, the agenda of propositions on which judgments are to be made is assumed to
contain at least two "atomic" propositions (e.g. P,
Q), one suitable "non-atomic" proposition (e.g. (P&Q))
and the negations of all these propositions.

Consider
three simple conditions on an aggregation procedure (informally stated):

All
of these papers consider independence conditions
weaker than systematicity (dropping the second part of systematicity,
which requires the same pattern of dependence to hold for all
propositions), but typically require agendas of propositions with richer logical
interconnections. For a critique of
the systematicity condition, see Chapman
(2002).

Among other results,
Pauly and van Hees show that the anonymity condition of the theorem can be relaxed to a
non-dictatorship condition; their paper also includes the first
impossibility theorem in the judgment aggregation literature in which
systematicity is weakened to independence.

Dietrich
and List (2005b) prove Arrow's theorem on preference aggregation as a
corollary of an impossibility theorem on judgment aggregation; for an
earlier comparison with impossibility results on preference aggregation,
see List
and Pettit (2001/2004). In both papers, preference orderings are
represented as sets of binary ranking propositions in predicate logic. Nehring
(2003) derives an Arrow-like theorem as a corollary of a theorem
in the property-space framework.

Dokow
and Holzman (2005) identify an algebraic condition on the structure of
logical interconnections between propositions that is necessary and
sufficient for an Arrow-style impossibility result; their paper also
develops connections between the judgment aggregation framework and
Wilson's (1975) as well as Rubinstein and Fishburn's (1986) frameworks.

Mongin
(2006) provides an impossibility result based on an independence
condition restricted to atomic propositions.

Dietrich
(2006) introduces a family of weakened independence conditions
capturing the idea that some (but not all) propositions are relevant
to certain others and proves several possibility and impossibility
results under these conditions..

Dietrich and List
(2004a)analyse strategic voting and strategy-proofness in judgment
aggregation. The
analysis is based on a non-preference-theoretic notion of
(non-)manipulability, but this notion is also related to a
preference-theoretic notion of strategy-proofness similar to the one
in standard social-choice-theoretic work on the Gibbard-Satterthwaite
theorem.

For
a model of judgment aggregation in general logics (where propositions can
be represented in several different logics, including propositional,
predicate, modal and conditional logics), see Dietrich
(2004).

Brennan
G. (2001) "Collective Coherence?" International Review of
Law and Economics 21(2): 197-211

Chapman,
B. (2001) "Public Reason, Social Choice, and Cooperation,"
paper presented at the Eighth Conference on Theoretical Aspects of
Rationality and Knowledge, University of Siena, held at Certosa di
Pontignano, Italy, July 2001 (PDF)
(source: http://chass.utoronto.ca/clea/confpapers.htm)

List, C., and P. Pettit
(2004)
"Aggregating Sets of Judgments: Two Impossibility Results
Compared," Synthese 140(1-2): 207-235; Australian National
University Working Paper in Social and Political Theory W20, 2001 (PDF)

Pigozzi,
G. (2005b) "Should we send him to prison? Paradoxes of aggregation
and belief merging," in We Will Show
Them: Essays in Honour of Dov Gabbay, Vol. 2, S. Artemov et al. (eds.), College
Publications: 529-542 (PDF)

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